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Difference between revisions of "2016 AMC 10A Problems/Problem 23"

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==Problem==
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== Problem ==
 
A binary operation <math>\diamondsuit</math> has the properties that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c</math> and that <math>a\,\diamondsuit \,a=1</math> for all nonzero real numbers <math>a, b,</math> and <math>c</math>. (Here <math>\cdot</math> represents multiplication). The solution to the equation <math>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p+q?</math>
 
A binary operation <math>\diamondsuit</math> has the properties that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c</math> and that <math>a\,\diamondsuit \,a=1</math> for all nonzero real numbers <math>a, b,</math> and <math>c</math>. (Here <math>\cdot</math> represents multiplication). The solution to the equation <math>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p+q?</math>
  
 
<math>\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601</math>
 
<math>\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601</math>
  
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==Solutions==
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== Solution 1 ==
  
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===Solution 1===
+
We see that <math>a \, \diamondsuit \, a = 1</math>, and think of division. Testing, we see that the first condition <math>a \, \diamondsuit \, (b \, \diamondsuit \, c) = (a \, \diamondsuit \, b) \cdot c</math> is satisfied, because <math>\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c</math>. Therefore, division can be the operation <math>\diamondsuit</math>. Solving the equation, <cmath>\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100\implies x=\frac{100}{336} = \frac{25}{84},</cmath> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109}</math>.
  
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We see that <math>a \diamond a = 1</math>, and think of division. Testing, we see that the first condition <math>a \diamond (b \diamond c) = (a \diamond b) \cdot c</math> is satisfied, because <math>\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c</math>. Therefore, division can be the operation <math>\diamond</math>. Solving the equation, <cmath>\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100\implies x=\frac{100}{336} = \frac{25}{84},</cmath> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109}</math>.
+
== Solution 2 (Proving that <math>\diamondsuit</math> is division) ==
 +
If the given conditions hold for all nonzero numbers <math>a, b,</math> and <math>c</math>,  
  
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===Solution 2===
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Let <math>a=b=c.</math> From the first two givens, this implies that
  
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We can manipulate the given identities to arrive at a conclusion about the binary operator <math>\diamondsuit</math>.  Substituting <math>b = c</math> into the first identity yields <cmath>( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\  b) = a\ \diamondsuit\  1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a.</cmath>  Hence, <math>( a\ \diamondsuit\ b) \cdot b = a,</math> or, dividing both sides of the equation by <math>b,</math> <math>( a\ \diamondsuit\ b) = \frac{a}{b}.</math>
 
  
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Hence, the given equation becomes <math>\frac{2016}{\frac{6}{x}} = 100</math>. Solving yields <math>x=\frac{100}{336} = \frac{25}{84},</math> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math>
+
<cmath>a\diamondsuit\, (a\diamondsuit\, {a})=(a\diamondsuit\, a)\cdot{a}.</cmath>
 +
 
 +
 
 +
From <math>a\diamondsuit\,{a}=1,</math> this equation simply becomes <math>a\diamondsuit\,{1}=a.</math>
 +
 
 +
 
 +
Let <math>c=b.</math> Substituting this into the first two conditions, we see that
 +
 
 +
<cmath>a\diamondsuit\, (b\diamondsuit\, {c})=(a\diamondsuit\, {b})\cdot{c} \implies a\diamondsuit\, (b\diamondsuit\, {b})=(a\diamondsuit\, {b})\cdot{b}.</cmath>
 +
 
 +
Substituting <math>b\diamondsuit\, {b} =1</math>, the second equation becomes
 +
 
 +
<cmath>a\diamondsuit\, {1}=(a\diamondsuit\, {b})\cdot{b} \implies a=(a\diamondsuit\,{b})\cdot{b}.</cmath>
 +
 
 +
Since <math>a, b</math> and <math>c</math> are nonzero, we can divide by <math>b</math> which yields,
 +
 
 +
<cmath>\frac{a}{b}=(a\diamondsuit\, {b}).</cmath>
 +
 
 +
Now we can find the value of <math>x</math> straightforwardly:
 +
 
 +
<cmath>\frac{2016}{(\frac{6}{x})}=100 \implies 2016=\frac{600}{x} \implies x=\frac{600}{2016} = \frac{25}{84}.</cmath>
 +
 
 +
Therefore, <math>a+b=25+84=\boxed{\textbf{A)} 109}</math>
 +
 
 +
-Benedict T (countmath1)
  
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===Solution 3===
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Note: We only really cared about what <math>a\diamondsuit\,{b}</math> was, so we used the existence of <math>c</math> to get an expression in terms of just <math>a</math> and <math>b</math>.
  
 +
== Solution 3 ==
 
One way to eliminate the <math>\diamondsuit</math> in this equation is to make <math>a = b</math> so that  <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = c</math>. In this case, we can make <math>b = 2016</math>.
 
One way to eliminate the <math>\diamondsuit</math> in this equation is to make <math>a = b</math> so that  <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = c</math>. In this case, we can make <math>b = 2016</math>.
  
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Therefore, <math>x = \frac{600}{2016} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math>
 
Therefore, <math>x = \frac{600}{2016} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math>
  
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==See Also==
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==Solution 4==
 +
 
 +
We can manipulate the given identities to arrive at a conclusion about the binary operator <math>\diamondsuit</math>.  Substituting <math>b = c</math> into the first identity yields <cmath>( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\  b) = a\ \diamondsuit\  1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a.</cmath>  Hence, <math>( a\ \diamondsuit\ b) \cdot b = a,</math> or, dividing both sides of the equation by <math>b,</math> <math>( a\ \diamondsuit\ b) = \frac{a}{b}.</math>
 +
 
 +
Hence, the given equation becomes <math>\frac{2016}{\frac{6}{x}} = 100</math>.  Solving yields <math>x=\frac{100}{336} = \frac{25}{84},</math> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math>
 +
 
 +
==Solution 5==
 +
 
 +
<math>2016 \diamondsuit (6 \diamondsuit x) = (2016 \diamondsuit 6) \cdot x = 100</math>
 +
 
 +
<math>2016 \diamondsuit (2016 \diamondsuit 1) = (2016 \diamondsuit 2016) \cdot 1 = 1 \cdot 1 = 1</math>
 +
 
 +
<math>2016 \diamondsuit 2016 = 1</math>, <math>2016 \diamondsuit (2016 \diamondsuit 1) = 1</math>, so <math>2016 \diamondsuit 1 = 2016</math>
 +
 
 +
<math>2016  \diamondsuit 1 = (2016  \diamondsuit 6) \cdot 6</math>, <math>2016 \diamondsuit 6 = \frac{2016  \diamondsuit 1}{6} = 336</math>
 +
 
 +
<math>x = \frac{100}{2016 \diamondsuit 6} = \frac{100}{336} = \frac{25}{84}</math>, <math>24 + 85 = \boxed{\textbf{(A) }109}</math>
 +
 
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 +
 
 +
==Solution 6 (Fast)==
 +
Notice that <math>2016 \diamondsuit (6 \diamondsuit 6)=(2016 \diamondsuit 6) \cdot 6 = 2016</math>. Hence, <math>2016 \diamondsuit 6 = 336</math>. Thus, <math>2016 \diamondsuit (6 \diamondsuit x)=100 \implies (2016 \diamondsuit 6) \cdot x = 100 \implies 336x=100 \implies x=\frac{25}{84}</math>. Therefore, the answer is <math>\boxed{\textbf{(A) }109}</math>
 +
 
 +
~Mathmagicops
 +
 
 +
==Solution 7==
 +
We can rewrite <math>2016\diamondsuit (6\diamondsuit x)=100</math> as <math>2016\diamondsuit 6=\dfrac{100}x.</math> We do a series of algebraic  manipulations:
 +
 
 +
<cmath>(2016\diamondsuit 6)\cdot6=\dfrac{600}x</cmath>
 +
<cmath>2016\diamondsuit (6\diamondsuit 6)=\dfrac{600}x</cmath>
 +
<cmath>2016\diamondsuit 1=\dfrac{600}x.</cmath>
 +
 
 +
Let <math>a=2016\diamondsuit 1.</math>
 +
 
 +
<cmath>2016a=(2016\diamondsuit 1)\cdot 2016</cmath>
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<cmath>2016a=2016\diamondsuit (1\diamondsuit 2016)</cmath>
 +
 
 +
We let <math>b=1\diamondsuit 2016.</math>
 +
 
 +
<cmath>2016b=(1\diamondsuit 2016)\cdot2016</cmath>
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<cmath>2016b=1\diamondsuit (2016\diamondsuit 2016)</cmath>
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<cmath>2016b=1\diamondsuit 1=1</cmath>
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<cmath>b=\dfrac1{2016}</cmath>
 +
 
 +
We notice that <math>1\diamondsuit 2016=\dfrac1{2016},</math> which makes <math>a\diamondsuit b</math> look suspiciously like <math>\dfrac ab.</math> Sure enough, when we try the given condition on <math>a\diamondsuit b=\dfrac ab,</math> we see that it works. We evaluate <math>2016\diamondsuit (6\diamondsuit x)=100,</math> and get <math>x=\dfrac{25}{84},</math> and therefore <math>p+q=25+84=109.</math>
 +
 
 +
~Technodoggo
 +
 
 +
== Video Solution 1==
 +
https://www.youtube.com/watch?v=8GULAMwu5oE
 +
 
 +
== Video Solution 2(Meta-Solving Technique) ==
 +
https://youtu.be/GmUWIXXf_uk?t=1632
 +
 
 +
~ pi_is_3.14
 +
 
 +
== See Also ==
 
{{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}}
 
{{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}}
 
{{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}}
 
{{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:43, 14 September 2023

Problem

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$

$\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

Solution 1

We see that $a \, \diamondsuit \, a = 1$, and think of division. Testing, we see that the first condition $a \, \diamondsuit \, (b \, \diamondsuit \, c) = (a \, \diamondsuit \, b) \cdot c$ is satisfied, because $\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c$. Therefore, division can be the operation $\diamondsuit$. Solving the equation, \[\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100\implies x=\frac{100}{336} = \frac{25}{84},\] so the answer is $25 + 84 = \boxed{\textbf{(A) }109}$.

Solution 2 (Proving that $\diamondsuit$ is division)

If the given conditions hold for all nonzero numbers $a, b,$ and $c$,

Let $a=b=c.$ From the first two givens, this implies that


\[a\diamondsuit\, (a\diamondsuit\, {a})=(a\diamondsuit\, a)\cdot{a}.\]


From $a\diamondsuit\,{a}=1,$ this equation simply becomes $a\diamondsuit\,{1}=a.$


Let $c=b.$ Substituting this into the first two conditions, we see that

\[a\diamondsuit\, (b\diamondsuit\, {c})=(a\diamondsuit\, {b})\cdot{c} \implies a\diamondsuit\, (b\diamondsuit\, {b})=(a\diamondsuit\, {b})\cdot{b}.\]

Substituting $b\diamondsuit\, {b} =1$, the second equation becomes

\[a\diamondsuit\, {1}=(a\diamondsuit\, {b})\cdot{b} \implies a=(a\diamondsuit\,{b})\cdot{b}.\]

Since $a, b$ and $c$ are nonzero, we can divide by $b$ which yields,

\[\frac{a}{b}=(a\diamondsuit\, {b}).\]

Now we can find the value of $x$ straightforwardly:

\[\frac{2016}{(\frac{6}{x})}=100 \implies 2016=\frac{600}{x} \implies x=\frac{600}{2016} = \frac{25}{84}.\]

Therefore, $a+b=25+84=\boxed{\textbf{A)} 109}$

-Benedict T (countmath1)

Note: We only really cared about what $a\diamondsuit\,{b}$ was, so we used the existence of $c$ to get an expression in terms of just $a$ and $b$.

Solution 3

One way to eliminate the $\diamondsuit$ in this equation is to make $a = b$ so that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = c$. In this case, we can make $b = 2016$.

\[2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100\implies (2016\, \diamondsuit\, 6) \cdot x = 100\]

By multiplying both sides by $\frac{6}{x}$, we get:

\[(2016\, \diamondsuit\, 6) \cdot 6 = \frac{600}{x}\implies 2016 \, \diamondsuit\, (6\, \diamondsuit\, 6) = \frac{600}{x}\]

Because $6\, \diamondsuit\, 6 = 2016\, \diamondsuit\, 2016 = 1:$

\[2016 \, \diamondsuit\, (2016\, \diamondsuit\, 2016) = \frac{600}{x}\implies (2016\, \diamondsuit\, 2016) \cdot 2016 = \frac{600}{x}\implies 2016 = \frac{600}{x}\]

Therefore, $x = \frac{600}{2016} = \frac{25}{84}$, so the answer is $25 + 84 = \boxed{\textbf{(A) }109.}$

Solution 4

We can manipulate the given identities to arrive at a conclusion about the binary operator $\diamondsuit$. Substituting $b = c$ into the first identity yields \[( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\ b) = a\ \diamondsuit\ 1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a.\] Hence, $( a\ \diamondsuit\ b) \cdot b = a,$ or, dividing both sides of the equation by $b,$ $( a\ \diamondsuit\ b) = \frac{a}{b}.$

Hence, the given equation becomes $\frac{2016}{\frac{6}{x}} = 100$. Solving yields $x=\frac{100}{336} = \frac{25}{84},$ so the answer is $25 + 84 = \boxed{\textbf{(A) }109.}$

Solution 5

$2016 \diamondsuit (6 \diamondsuit x) = (2016 \diamondsuit 6) \cdot x = 100$

$2016 \diamondsuit (2016 \diamondsuit 1) = (2016 \diamondsuit 2016) \cdot 1 = 1 \cdot 1 = 1$

$2016 \diamondsuit 2016 = 1$, $2016 \diamondsuit (2016 \diamondsuit 1) = 1$, so $2016 \diamondsuit 1 = 2016$

$2016 \diamondsuit 1 = (2016 \diamondsuit 6) \cdot 6$, $2016 \diamondsuit 6 = \frac{2016 \diamondsuit 1}{6} = 336$

$x = \frac{100}{2016 \diamondsuit 6} = \frac{100}{336} = \frac{25}{84}$, $24 + 85 = \boxed{\textbf{(A) }109}$

~isabelchen

Solution 6 (Fast)

Notice that $2016 \diamondsuit (6 \diamondsuit 6)=(2016 \diamondsuit 6) \cdot 6 = 2016$. Hence, $2016 \diamondsuit 6 = 336$. Thus, $2016 \diamondsuit (6 \diamondsuit x)=100 \implies (2016 \diamondsuit 6) \cdot x = 100 \implies 336x=100 \implies x=\frac{25}{84}$. Therefore, the answer is $\boxed{\textbf{(A) }109}$

~Mathmagicops

Solution 7

We can rewrite $2016\diamondsuit (6\diamondsuit x)=100$ as $2016\diamondsuit 6=\dfrac{100}x.$ We do a series of algebraic manipulations:

\[(2016\diamondsuit 6)\cdot6=\dfrac{600}x\] \[2016\diamondsuit (6\diamondsuit 6)=\dfrac{600}x\] \[2016\diamondsuit 1=\dfrac{600}x.\]

Let $a=2016\diamondsuit 1.$

\[2016a=(2016\diamondsuit 1)\cdot 2016\] \[2016a=2016\diamondsuit (1\diamondsuit 2016)\]

We let $b=1\diamondsuit 2016.$

\[2016b=(1\diamondsuit 2016)\cdot2016\] \[2016b=1\diamondsuit (2016\diamondsuit 2016)\] \[2016b=1\diamondsuit 1=1\] \[b=\dfrac1{2016}\]

We notice that $1\diamondsuit 2016=\dfrac1{2016},$ which makes $a\diamondsuit b$ look suspiciously like $\dfrac ab.$ Sure enough, when we try the given condition on $a\diamondsuit b=\dfrac ab,$ we see that it works. We evaluate $2016\diamondsuit (6\diamondsuit x)=100,$ and get $x=\dfrac{25}{84},$ and therefore $p+q=25+84=109.$

~Technodoggo

Video Solution 1

https://www.youtube.com/watch?v=8GULAMwu5oE

Video Solution 2(Meta-Solving Technique)

https://youtu.be/GmUWIXXf_uk?t=1632

~ pi_is_3.14

See Also

2016 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2016 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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